Solution of dirichlet problem in terms of elliptic functions

vıı ABSTRACT In this study, the Green function of the Dirichlet problem is defined in terms of elliptic functions and the solution of the problem is based on the Green function and thus elliptic functions. In the first chapter, the history and development of differential equations, the history of elliptic functions, the definition of Dirichlet problem and previous studies are presented. In the second chapter, fundamental relations such as elliptic functions and related operations, elliptic integrals, conform mapping, Riemann transformation theorem and Green theorem are mentioned. In the third chapter, methods are given for the solution of the problem. In this chapter, the method of solution for Dirichlet problem by Green function is presented first and then the Green function and the transformation required for the solution of the Dirichlet problem in the rectangular region is found and the problem is solved in the rectangular region. Subsequently, the Green function and the transformation required for the Dirichlet problem in the ellipse region is found and the problem is solved in the ellipse region. In the fourth chapter, an application for the problem in the rectangular region is given in order to present an application of the solution of Dirichlet problem in terms of elliptic functions. At first, the equation is solved in terms of elliptic functions. Then the same equation is solved by the known Method of Separation of Variables, and the results of both methods are compared. We made use Mathcad computer program to make the comparison. The consistency of these sensitive results are shown by graphics. In the fifth chapter, results and discussion are given, the facilities and difficulties of the method presented and the contributions of the method are mentioned.